Catalan numbers, parking functions, and invariant theory

Size: px
Start display at page:

Download "Catalan numbers, parking functions, and invariant theory"

Transcription

1 Catalan numbers, parking functions, and invariant theory Vic Reiner Univ. of Minnesota CanaDAM Memorial University, Newfoundland June 10, 2013

2 Outline 1 Catalan numbers and objects 2 Parking functions and parking space (type A) 3 q-catalan numbers and cyclic symmetry 4 Reflection group generalization

3 Catalan numbers Definition The Catalan number is Cat n := 1 ( ) 2n n+1 n Example Cat 3 = 1 4 ( ) 6 = 5. 3 It s not even completely obvious it is always an integer. But it counts many things, at least 205, as of June 6, 2013, according to Richard Stanley s Catalan addendum.

4 Catalan numbers Definition The Catalan number is Cat n := 1 ( ) 2n n+1 n Example Cat 3 = 1 4 ( ) 6 = 5. 3 It s not even completely obvious it is always an integer. But it counts many things, at least 205, as of June 6, 2013, according to Richard Stanley s Catalan addendum. Let s recall a few of them.

5 Triangulations of an (n + 2)-gon Example There are 5 = Cat 3 triangulations of a pentagon.

6 Catalan paths Definition A Catalan path from (0, 0) to (n, n) is a path taking unit north or east steps staying weakly below y = x. Example The are 5 = Cat 3 Catalan paths from (0, 0) to (3, 3).

7 Increasing parking functions Definition An increasing parking function of size n is an integer sequence (a 1, a 2,..., a n ) with 1 a i i. They give the heights of horizontal steps in Catalan paths. Example

8 Nonnesting and noncrossing partitions of {1, 2,..., n} Example nesting: nonnesting:

9 Nonnesting and noncrossing partitions of {1, 2,..., n} Example nesting: nonnesting: Example crossing: noncrossing:

10 Nonnesting partitions NN(3) of {1, 2, 3} Example There are 5 = Cat 3 nonnesting partitions of {1, 2, 3}

11 Noncrossing partitions NC(3) of {1, 2, 3} Example There are 5 = Cat 3 noncrossing partitions of {1, 2, 3}

12 NN(4) versus NC(4) is slightly more interesting Example For n = 4, among the 15 set partitions of {1, 2, 3, 4}, exactly one is nesting, and exactly one is crossing, leaving 14 = Cat 4 nonnesting or noncrossing partitions.

13 So what are the parking functions? Definition Parking functions of length n are sequences (f(1),..., f(n)) for which f 1 ({1, 2,..., i}) i for i = 1, 2,..., n. Definition (The cheater s version) Parking functions of length n are sequences (f(1),..., f(n)) whose weakly increasing rearrangement is an increasing parking function!

14 The parking function number (n+1) n 1 Theorem (Konheim and Weiss 1966) There are (n+1) n 1 parking functions of length n. Example For n = 3, the (3+1) 3 1 = 16 parking functions of length 3, grouped by their increasing parking function rearrangement, leftmost:

15 Parking functions as coset representatives Proposition (Haiman 1993) The (n+1) n 1 parking functions give coset representatives for Z n /(Z[1, 1,...,1]+(n+1)Z n )

16 Parking functions as coset representatives Proposition (Haiman 1993) The (n+1) n 1 parking functions give coset representatives for Z n /(Z[1, 1,...,1]+(n+1)Z n ) or equivalently, by a Noether isomorphism theorem, for (Z n+1 ) n /Z n+1 [1, 1,..., 1]

17 Parking functions as coset representatives Proposition (Haiman 1993) The (n+1) n 1 parking functions give coset representatives for Z n /(Z[1, 1,...,1]+(n+1)Z n ) or equivalently, by a Noether isomorphism theorem, for (Z n+1 ) n /Z n+1 [1, 1,..., 1] or equivalently, by the same isomorphism theorem, for Q/(n+1)Q where here Q is the rank n 1 lattice Q := Z n /Z[1, 1,..., 1] = Z n 1.

18 So what s the parking space? The parking space is the permutation representation of W = S n, acting on the (n+1) n 1 parking functions of length n. Example For n = 3 it is the permutation representation of W = S 3 on these words, with these orbits:

19 Wondrous! Just about every natural question about this W-permutation representation Park n has a beautiful answer. Many were noted by Haiman in his 1993 paper Conjectures on diagonal harmonics.

20 Wondrous! Just about every natural question about this W-permutation representation Park n has a beautiful answer. Many were noted by Haiman in his 1993 paper Conjectures on diagonal harmonics. As the parking functions give coset representatives for the quotient Q/(n+1)Q where Q := Z n /Z[1, 1,...,1] = Z n 1, one can deduce this. Corollary Each permutation w in W = S n acts on Park n with character value = trace = number of fixed parking functions χ Parkn (w) = (n+ 1) #(cycles of w) 1.

21 Orbit structure? We ve seen the W-orbits in Park n are parametrized by increasing parking functions, which are Catalan objects. The stabilizer of an orbit is always a Young subgroup S λ := S λ1 S λl where λ are the multiplicities in any orbit representative. Example 111 (3) (2,1) (2,1) (2,1) (1,1,1) λ

22 Orbit structure via the nonnesting or noncrossing partitions That same stabilizer data S λ is predicted by the block sizes in nonnesting partitions, or noncrossing partitions of {1, 2,..., n}.

23 Nonnesting partitions NN(3) of {1, 2, 3} (3) (2, 1) (2, 1) (2, 1) (1, 1, 1) Theorem (Shi 1986, Cellini-Papi 2002) NN(n) bijects to increasing parking functions respecting λ.

24 Noncrossing partitions NC(3) of {1, 2, 3} (3) (2, 1) (2, 1) (2, 1) (1, 1, 1) Theorem (Athanasiadis 1998) There is a bijection NN(n) NC(n), respecting λ.

25 The block size equidistribution for NN(4) versus NC(4) Example Recall that among the 15 set partitions of {1, 2, 3, 4}, exactly one was nesting, and exactly one was crossing, and note that both correspond to λ = (2, 2).

26 More wonders: Irreducible multiplicities in Park n For W = S n, the irreducible characters are {χ λ } indexed by partitions λ of n. Haiman gave a product formula for any of the irreducible multiplicities χ λ, Park n.

27 More wonders: Irreducible multiplicities in Park n For W = S n, the irreducible characters are {χ λ } indexed by partitions λ of n. Haiman gave a product formula for any of the irreducible multiplicities χ λ, Park n. The special case of hook shapes λ = (n k, 1 k ) becomes this. Theorem (Pak-Postnikov 1997) The multiplicity χ (n k,1k),χ Parkn W is the number of subdivisions of an (n+2)-gon using n 1 k internal diagonals, or the number of k-dimensional faces in the (n 1)-dimensional associahedron.

28 Example: n=4 χ (3),χ Park3 S3 = 5 χ (2,1),χ Park3 S3 = 5 χ (1,1,1),χ Park3 S3 = 1 χ (4),χ Park4 S4 = 14 χ (3,1),χ Park4 S4 = 21 χ (2,1,1),χ Park4 S4 = 9 χ (1,1,1,1),χ Park4 S4 = 1

29 q-catalan numbers Let s rewrite the Catalan number as Cat n = 1 ( ) 2n = (n+2)(n+3) (2n) n+1 n (2)(3) (n)

30 q-catalan numbers Let s rewrite the Catalan number as Cat n = 1 ( ) 2n = (n+2)(n+3) (2n) n+1 n (2)(3) (n) and consider MacMahon s q-catalan number 1 Cat n (q) = [n+1] q [ ] 2n n q := (1 qn+2 )(1 q n+3 ) (1 q 2n ) (1 q 2 )(1 q 3 ) (1 q n. )

31 The q-catalan hides information on cyclic symmetries The noncrossings NC(n) have a Z/nZ-action via rotations, whose orbit structure is completely predicted by root-of-unity evaluations of this q-catalan number.

32 The q-catalan hides information on cyclic symmetries The noncrossings NC(n) have a Z/nZ-action via rotations, whose orbit structure is completely predicted by root-of-unity evaluations of this q-catalan number. Theorem (Stanton-White-R. 2004) For d dividing n, the number of noncrossing partitions of n with d-fold rotational symmetry is [Cat n (q)] q=ζd where ζ d is any primitive d th root of unity in C. We called such a set-up a cyclic sieving phenomenon.

33 NC(4), Cat 4 (q) and rotational symmetry Example Via L Hôpital s rule, for example, one can evaluate Cat 4 (q) = (1 q6 )(1 q 7 )(1 q 8 ) 14 if q = +1 = ζ 1 (1 q 2 )(1 q 3 )(1 q 4 ) = 6 if q = 1 = ζ 2 2 if q = ±i = ζ 4.

34 NC(4), Cat 4 (q) and rotational symmetry Example Via L Hôpital s rule, for example, one can evaluate Cat 4 (q) = (1 q6 )(1 q 7 )(1 q 8 ) 14 if q = +1 = ζ 1 (1 q 2 )(1 q 3 )(1 q 4 ) = 6 if q = 1 = ζ 2 2 if q = ±i = ζ 4. predicting 14 elements of NC(4) total,

35 NC(4), Cat 4 (q) and rotational symmetry Example Via L Hôpital s rule, for example, one can evaluate Cat 4 (q) = (1 q6 )(1 q 7 )(1 q 8 ) 14 if q = +1 = ζ 1 (1 q 2 )(1 q 3 )(1 q 4 ) = 6 if q = 1 = ζ 2 2 if q = ±i = ζ 4. predicting 14 elements of NC(4) total, 6 with 2-fold symmetry,

36 NC(4), Cat 4 (q) and rotational symmetry Example Via L Hôpital s rule, for example, one can evaluate Cat 4 (q) = (1 q6 )(1 q 7 )(1 q 8 ) 14 if q = +1 = ζ 1 (1 q 2 )(1 q 3 )(1 q 4 ) = 6 if q = 1 = ζ 2 2 if q = ±i = ζ 4. predicting 14 elements of NC(4) total, 6 with 2-fold symmetry, of which have 4-fold rotational symmetry.

37 Cat n (q) does double duty hiding cyclic orbit data Definition For a finite poset P, the Duchet-FonDerFlaass (rowmotion) cyclic action maps an antichain A Ψ(A) to the minimal elementsψ(a) among elements below no element of A. That is, Example Ψ(A) := min{p \ P A }. In P the (3, 2, 1) staircase poset, one has A = Ψ(A) = 7 7 7

38 The Ψ-orbits for the staircase poset (3, 2, 1) There is a size 2 orbit:

39 The Ψ-orbits for the staircase poset (3, 2, 1) There is a size 2 orbit: A size 4 orbit (= the rank sets of the poset, plus A = ):

40 The Ψ-orbits for the staircase poset (3, 2, 1) There is a size 2 orbit: A size 4 orbit (= the rank sets of the poset, plus A = ): A size 8 orbit:

41 Cat n (q) is doing double duty Theorem (part of Armstrong-Stump-Thomas 2011) For d dividing 2n (not n this time), the number of antichains in the (n 1, n 2,..., 2, 1) staircase poset fixed by Ψ d is [Cat n (q)] q=ζd (And these antichains are really disguised Catalan paths.) Example

42 How did their theorem predict those orbit sizes? Example For n = 4 it predicted that, of the 14 = Cat 4 antichains, we d see Cat 4 (q) = (1 q6 )(1 q 7 )(1 q 8 ) (1 q 2 )(1 q 3 )(1 q 4 ) 14 fixed by Ψ 8 from setting q = +1 = ζ 1 6 fixed by Ψ 4 from setting q = 1 = ζ 2 = 2 fixed by Ψ 2 from setting q = i = ζ 4 0 fixed by Ψ 1 from setting q = e πi 4 = ζ 8.

43 How did their theorem predict those orbit sizes? Example For n = 4 it predicted that, of the 14 = Cat 4 antichains, we d see Cat 4 (q) = (1 q6 )(1 q 7 )(1 q 8 ) (1 q 2 )(1 q 3 )(1 q 4 ) 14 fixed by Ψ 8 from setting q = +1 = ζ 1 6 fixed by Ψ 4 from setting q = 1 = ζ 2 = 2 fixed by Ψ 2 from setting q = i = ζ 4 0 fixed by Ψ 1 from setting q = e πi 4 = ζ 8. This means there are no singleton orbits, one orbit of size 2, one of size 4 = 6 2, and one orbit of size 8 = 14 6, that is, one free orbit.

44 Actually Cat n (q) is doing triple duty! Theorem (Stanton-White-R. 2004) For d dividing n + 2, the number of d-fold rotationally symmetric triangulations of an (n+ 2)-gon is [Cat n (q)] q=ζd Example For n = 4, these rotation orbit sizes for triangulations of a hexagon are predicted by 14 if q = +1 = ζ 1 Cat 4 (q) = (1 q6 )(1 q 7 )(1 q 8 ) (1 q 2 )(1 q 3 )(1 q 4 ) = 6 if q = 1 = ζ 2 2 if q = e 2πi3 = ζ 3 0 if q = e 2πi6 = ζ 6

45 On to the reflection group generalizations Generalize to irreducible real ref n groups W acting on V = R n. Example W = S n acts irreducibly on V = R n 1, realized as x 1 + x 2 + +x n = 0 within R n. It is generated transpositions (i, j), which are reflections through the hyperplanes x i = x j s 2 s 1 s

46 Invariant theory enters the picture Theorem (Chevalley, Shephard-Todd 1955) When W acts on polynomials S = C[x 1,...,x n ] = Sym(V ), its W-invariant subalgebra is again a polynomial algebra S W = C[f 1,..., f n ] One can pick f 1,...,f n homogeneous, with degrees d 1 d 2 d n, and define h := d n the Coxeter number.

47 Invariant theory enters the picture Theorem (Chevalley, Shephard-Todd 1955) When W acts on polynomials S = C[x 1,...,x n ] = Sym(V ), its W-invariant subalgebra is again a polynomial algebra S W = C[f 1,..., f n ] One can pick f 1,...,f n homogeneous, with degrees d 1 d 2 d n, and define h := d n the Coxeter number. Example For W = S n, one has S W = C[e 2 (x),..., e n (x)], so the degrees are (2, 3,...,n), and h = n.

48 Weyl groups and the first W-parking space When W is a Weyl (crystallographic) real finite reflection group, it preserves a full rank lattice Q = Z n inside V = R n. One can choose a root system Φ of normals to the hyperplanes, in such a way that the root lattice Q := ZΦ is a W-stable lattice.

49 Weyl groups and the first W-parking space When W is a Weyl (crystallographic) real finite reflection group, it preserves a full rank lattice Q = Z n inside V = R n. One can choose a root system Φ of normals to the hyperplanes, in such a way that the root lattice Q := ZΦ is a W-stable lattice. Definition (Haiman 1993) We should think of the W-permutation representation on the set Park(W) := Q/(h+1)Q as a W-analogue of parking functions.

50 Wondrous properties of Park(w) = Q/(h + 1)Q Theorem (Haiman 1993) For a Weyl group W, #Q/(h+1)Q = (h+1) n.

51 Wondrous properties of Park(w) = Q/(h + 1)Q Theorem (Haiman 1993) For a Weyl group W, #Q/(h+1)Q = (h+1) n. Any w in W acts with trace (character value) χ Park(W) (w) = (h+1) dim V w.

52 Wondrous properties of Park(w) = Q/(h + 1)Q Theorem (Haiman 1993) For a Weyl group W, #Q/(h+1)Q = (h+1) n. Any w in W acts with trace (character value) χ Park(W) (w) = (h+1) dim V w. The W-orbit count #W\Q/(h + 1)Q is the W-Catalan: 1 W,χ Park(W) = n i=1 h+d i d i =: Cat(W)

53 W-Catalan example: W = S n Example Recall that W = S n acts irreducibly on V = R n 1 with degrees (2, 3,...,n) and h = n. One can identify the root lattice Q = Z n /(1, 1,..., 1)Z. One has #Q/(h+1)Q = (n+1) n 1, and Cat(S n ) = #W\Q/(h + 1)Q = (n+2)(n+3) (2n) 2 3 n = 1 ( ) 2n n+1 n = Cat n.

54 Exterior powers of V One can consider multiplicities in Park(W) not just of 1 W = 0 V det W = n V but all the exterior powers k V for k = 0, 1, 2,...,n, which are known to all be W-irreducibles (Steinberg). Example W = S n acts irreducibly on V = R n 1 with character χ (n 1,1), and on k V with character χ (n k,1k).

55 Theorem (Armstrong-Rhoades-R. 2012) For Weyl groups W, the multiplicity χ k V,χ Park(W) is the number of (n k)-element sets of compatible cluster variables in a cluster algebra of finite type W, or the number of k-dimensional faces in the W-associahedron of Chapoton-Fomin-Zelevinsky (2002).

56 Two W-Catalan objects: NN(W) and NC(W) The previous result relies on an amazing coincidence for two W-Catalan counted families generalizing NN(n), NC(n). Definition (Postnikov 1997) For Weyl groups W, define W-nonnesting partitions NN(W) to be the antichains in the poset of positive roots Φ +. Example corresponds to this antichain A: e 1 e 5 e 1 e 4 e 2 e 5 e 1 e 3 e 2 e 4 e 3 e 5 e 1 e 2 e 2 e 3 e 3 e 4 e 4 e 5

57 W-noncrossing partitions Definition (Bessis 2003, Brady-Watt 2002) W-noncrossing partitions NC(W) are the interval [e, c] abs from identity e to any Coxeter element c in absolute order abs on W: x abs y if l T (x)+l T (x 1 y) = l T (y) where the absolute (reflection) length is l T (w) = min{w = t 1 t 2 t l : t i reflections} and a Coxeter element c = s 1 s 2 s n is any product of a choice of simple reflections S = {s 1,...,s n } s 2 s 1 s

58 The case W = S n Example For W = S n, the n-cycle c = (1, 2,..., n) is one choice of a Coxeter element. And permutations w in NC(W) = [e, c] abs come from orienting clockwise the blocks of the noncrossing partitions NC(n)

59 The absolute order on W = S 3 and NC(S 3 ) Example

60 Generalizing NN, NC block size coincidence We understand why NN(W) is counted by Cat(W). We do not really understand why the same holds for NC(W). Worse, we do not really understand why the following holds it was checked case-by-case. Theorem (Athanasiadis-R. 2004) The W-orbit distributions coincide a for subspaces arising as intersections X = α A α for A in NN(W), and as fixed spaces X = V w for w in NC(W). a...and have a nice product formula via Orlik-Solomon exponents.

61 What about a q-analogue of Cat(W)? Theorem (Gordon 2002, Berest-Etingof-Ginzburg 2003) For irreducible real reflection groups W, Cat(W, q) := n i=1 1 q h+d i 1 q d i turns out to lie in N[q], as it is a Hilbert series Cat(W, q) = Hilb( (S/(Θ)) W, q) where Θ = (θ 1,...,θ n ) is a magical hsop in S = C[x 1,...,x n ] Here magical means...

62 What about a q-analogue of Cat(W)? Theorem (Gordon 2002, Berest-Etingof-Ginzburg 2003) For irreducible real reflection groups W, Cat(W, q) := n i=1 1 q h+d i 1 q d i turns out to lie in N[q], as it is a Hilbert series Cat(W, q) = Hilb( (S/(Θ)) W, q) where Θ = (θ 1,...,θ n ) is a magical hsop in S = C[x 1,...,x n ] Here magical means... (θ 1,...,θ n ) are homogeneous, all of degree h+1,

63 What about a q-analogue of Cat(W)? Theorem (Gordon 2002, Berest-Etingof-Ginzburg 2003) For irreducible real reflection groups W, Cat(W, q) := n i=1 1 q h+d i 1 q d i turns out to lie in N[q], as it is a Hilbert series Cat(W, q) = Hilb( (S/(Θ)) W, q) where Θ = (θ 1,...,θ n ) is a magical hsop in S = C[x 1,...,x n ] Here magical means... (θ 1,...,θ n ) are homogeneous, all of degree h+1, their C-span carries W-rep n V, like {x 1,..., x n }, and

64 What about a q-analogue of Cat(W)? Theorem (Gordon 2002, Berest-Etingof-Ginzburg 2003) For irreducible real reflection groups W, Cat(W, q) := n i=1 1 q h+d i 1 q d i turns out to lie in N[q], as it is a Hilbert series Cat(W, q) = Hilb( (S/(Θ)) W, q) where Θ = (θ 1,...,θ n ) is a magical hsop in S = C[x 1,...,x n ] Here magical means... (θ 1,...,θ n ) are homogeneous, all of degree h+1, their C-span carries W-rep n V, like {x 1,..., x n }, and S/(Θ) is finite-dim l (=: the graded W-parking space).

65 Do you believe in magic? These magical hsop s do exist, and they re not unique. Example For W = B n, the hyperoctahedral group of signed permutation matrices, acting on V = R n, one has h = 2n, and one can take Example Θ = (x 2n+1 1,..., x 2n+1 n ). For W = S n they re tricky. A construction by Kraft appears in Haiman (1993), and Dunkl (1998) gave another. For general real reflection groups, Θ comes from rep theory of the rational Cherednik algebra for W, with parameter h+1 h.

66 Cat(W, q) and cyclic symmetry Cat(W, q) interacts well with a cyclic Z/hZ-action on NC(W) = [e, c] abs that comes from conjugation w cwc 1, generalizing rotation of noncrossing partitions NC(n). Theorem (Bessis-R. 2004) For any d dividing h, the number of w in NC(W) that have d-fold symmetry, meaning that c h d wc h d = w, is [Cat(W, q)] q=ζd where ζ d is any primitive d th root of unity in C.

67 Cat(W, q) and cyclic symmetry Cat(W, q) interacts well with a cyclic Z/hZ-action on NC(W) = [e, c] abs that comes from conjugation w cwc 1, generalizing rotation of noncrossing partitions NC(n). Theorem (Bessis-R. 2004) For any d dividing h, the number of w in NC(W) that have d-fold symmetry, meaning that c h d wc h d = w, is [Cat(W, q)] q=ζd where ζ d is any primitive d th root of unity in C. But the proof again needed some of the case-by-case facts!

68 Cat(W, q) does double duty Generalizing behavior of A Ψ(A) in the staircase posets, Armstrong, Stump and Thomas (2011) actually proved the following general statement, conjectured in Bessis-R. (2004), suggested by weaker conjectures of Panyushev (2007). Theorem (Armstrong-Stump-Thomas 2011) For Weyl group W, and for d dividing 2h (not h this time), the number of antichains in the positive root posetφ + fixed by Ψ d is [Cat(W, q)] q=ζd A = Ψ(A) = 7 7 7

69 Cat(W, q) does double duty Generalizing behavior of A Ψ(A) in the staircase posets, Armstrong, Stump and Thomas (2011) actually proved the following general statement, conjectured in Bessis-R. (2004), suggested by weaker conjectures of Panyushev (2007). Theorem (Armstrong-Stump-Thomas 2011) For Weyl group W, and for d dividing 2h (not h this time), the number of antichains in the positive root posetφ + fixed by Ψ d is [Cat(W, q)] q=ζd A = Ψ(A) = Again, part of the arguments rely on case-by-case verifications.

70 Cat(W, q) does triple duty Generalizing what happens for rotating triangulations of polygons, Eu and Fu proved the following statement that we had conjectured. Theorem (Eu and Fu 2011) For Weyl group W, and for d dividing h+2 (not h, nor 2h this time), the number of clusters having d-fold symmetry under Fomin and Zelevinsky s deformed Coxeter element is [Cat(W, q)] q=ζd

71 Cat(W, q) does triple duty Again, part ofvic the Reiner arguments Univ. of Minnesota rely Catalan on case-by-case numbers, parking functions, verifications. and invariant theory Generalizing what happens for rotating triangulations of polygons, Eu and Fu proved the following statement that we had conjectured. Theorem (Eu and Fu 2011) For Weyl group W, and for d dividing h+2 (not h, nor 2h this time), the number of clusters having d-fold symmetry under Fomin and Zelevinsky s deformed Coxeter element is [Cat(W, q)] q=ζd

72 The big question Question Can we get rid of the case-by-case, and really understand why these things hold so generally?

73 The big question Question Can we get rid of the case-by-case, and really understand why these things hold so generally? Thanks for listening!

Rational Catalan Combinatorics: Intro

Rational Catalan Combinatorics: Intro Rational Catalan Combinatorics: Intro Vic Reiner Univ. of Minnesota reiner@math.umn.edu AIM workshop Dec. 17-21, 2012 Goals of the workshop 1 Reinforce existing connections and forge new connections between

More information

Noncrossing Parking Functions

Noncrossing Parking Functions Noncrossing Parking Functions Drew Armstrong (with B. Rhoades and V. Reiner) University of Miami www.math.miami.edu/ armstrong Non-crossing partitions in representation theory Bielefeld, June 2014 Plan

More information

Counting chains in noncrossing partition lattices

Counting chains in noncrossing partition lattices Counting chains in noncrossing partition lattices Nathan Reading NC State University NCSU Algebra Seminar, November 16, 2007 1 Counting chains in noncrossing partition lattices Classical noncrossing partitions

More information

arxiv: v2 [math.co] 9 May 2012

arxiv: v2 [math.co] 9 May 2012 PARKING SPACES DREW ARMSTRONG, VICTOR REINER, AND BRENDON RHOADES arxiv:1204.1760v2 [math.co] 9 May 2012 Abstract. Let W be a Weyl group with root lattice Q and Coxeter number h. The elements of the finite

More information

A cyclic sieving phenomenon in Catalan Combinatorics

A cyclic sieving phenomenon in Catalan Combinatorics A cyclic sieving phenomenon in Catalan Combinatorics Christian Stump Université du Québec à Montréal SLC 64, Lyon March 30, 2010 Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM,

More information

Rational Catalan combinatorics

Rational Catalan combinatorics Rational Catalan combinatorics organized by Drew Armstrong, Stephen Griffeth, Victor Reiner, and Monica Vazirani Workshop Summary 1. Topics of the Workshop 1 The mathematics associated with the classical

More information

Outline. Some Reflection Group Numerology. Root Systems and Reflection Groups. Example: Symmetries of a triangle. Paul Renteln

Outline. Some Reflection Group Numerology. Root Systems and Reflection Groups. Example: Symmetries of a triangle. Paul Renteln Outline 1 California State University San Bernardino and Caltech 2 Queen Mary University of London June 13, 2014 3 Root Systems and Reflection Groups Example: Symmetries of a triangle V an n dimensional

More information

Hidden Symmetries of the Root Systems

Hidden Symmetries of the Root Systems Universität Bielefeld, SJTU Shanghai, KAU Jeddah 57th Annual Meeting of the Australian Mathematical Society Sydney, October 2, 2013 Symmetry is one idea by which man through the ages has tried to comprehend

More information

Torus Knots and q, t-catalan Numbers

Torus Knots and q, t-catalan Numbers Torus Knots and q, t-catalan Numbers Eugene Gorsky Stony Brook University Simons Center For Geometry and Physics April 11, 2012 Outline q, t-catalan numbers Compactified Jacobians Arc spaces on singular

More information

A BIJECTION BETWEEN NONCROSSING AND NONNESTING PARTITIONS OF TYPES A, B AND C

A BIJECTION BETWEEN NONCROSSING AND NONNESTING PARTITIONS OF TYPES A, B AND C Volume 6, Number 2, Pages 70 90 ISSN 1715-0868 A BIJECTION BETWEEN NONCROSSING AND NONNESTING PARTITIONS OF TYPES A, B AND C RICARDO MAMEDE Abstract The total number of noncrossing ( partitions of type

More information

A Survey of Parking Functions

A Survey of Parking Functions A Survey of Parking Functions Richard P. Stanley M.I.T. Parking functions... n 2 1... a a a 1 2 n Car C i prefers space a i. If a i is occupied, then C i takes the next available space. We call (a 1,...,a

More information

Outline. The Distance Spectra of Cayley Graphs of Coxeter Groups. Finite Reflection Group. Root Systems. Reflection/Coxeter Groups.

Outline. The Distance Spectra of Cayley Graphs of Coxeter Groups. Finite Reflection Group. Root Systems. Reflection/Coxeter Groups. The Distance Spectra of of Coxeter Groups Paul Renteln 1 Department of Physics California State University San Bernardino 2 Department of Mathematics California Institute of Technology ECNU, Shanghai June

More information

NOTES ON POINCARÉ SERIES OF FINITE AND AFFINE COXETER GROUPS

NOTES ON POINCARÉ SERIES OF FINITE AND AFFINE COXETER GROUPS NOTES ON POINCARÉ SERIES OF FINITE AND AFFINE COXETER GROUPS VICTOR REINER Abstract. There are two famous formulae relating the Poincaré series of a finite/affine Weyl group to the degrees of fundamental

More information

Factorisations of the Garside element in the dual braid monoids

Factorisations of the Garside element in the dual braid monoids Factorisations of the Garside element in the dual braid monoids Vivien Ripoll École Normale Supérieure Département de Mathématiques et Applications 30 June 2010 Journées Garside Caen 1 Dual braid monoids

More information

A BIJECTION BETWEEN NONCROSSING AND NONNESTING PARTITIONS FOR CLASSICAL REFLECTION GROUPS. Senior Thesis. Benjamin Iriarte Giraldo

A BIJECTION BETWEEN NONCROSSING AND NONNESTING PARTITIONS FOR CLASSICAL REFLECTION GROUPS. Senior Thesis. Benjamin Iriarte Giraldo A BIJECTION BETWEEN NONCROSSING AND NONNESTING PARTITIONS FOR CLASSICAL REFLECTION GROUPS Senior Thesis Benjamin Iriarte Giraldo THESIS ADVISOR: Federico Ardila Author address: Pregrado Universitario,

More information

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS ANA BALIBANU DISCUSSED WITH PROFESSOR VICTOR GINZBURG 1. Introduction The aim of this paper is to explore the geometry of a Lie algebra g through the action

More information

On the H-triangle of generalised nonnesting partitions

On the H-triangle of generalised nonnesting partitions FPSAC 204, Chicago, USA DMTCS proc. AT, 204, 83 90 On the H-triangle of generalised nonnesting partitions Marko Thiel Department of Mathematics, University of Vienna, Austria Abstract. With a crystallographic

More information

CHRISTOS A. ATHANASIADIS, THOMAS BRADY, AND COLUM WATT

CHRISTOS A. ATHANASIADIS, THOMAS BRADY, AND COLUM WATT SHELLABILITY OF NONCROSSING PARTITION LATTICES CHRISTOS A. ATHANASIADIS, THOMAS BRADY, AND COLUM WATT Abstract. We give a case-free proof that the lattice of noncrossing partitions associated to any finite

More information

The Waring rank of the Vandermonde determinant

The Waring rank of the Vandermonde determinant The Waring rank of the Vandermonde determinant Alexander Woo (U. Idaho) joint work with Zach Teitler(Boise State) SIAM Conference on Applied Algebraic Geometry, August 3, 2014 Waring rank Given a polynomial

More information

Hyperplane Arrangements & Diagonal Harmonics

Hyperplane Arrangements & Diagonal Harmonics Hyperplane Arrangements & Diagonal Harmonics Drew Armstrong arxiv:1005.1949 Coinvariants Coinvariants Theorems (Newton-Chevalley-etc): Let S n act on S = C[x 1,..., x n ] by permuting variables. Coinvariants

More information

Euler characteristic of the truncated order complex of generalized noncrossing partitions

Euler characteristic of the truncated order complex of generalized noncrossing partitions Euler characteristic of the truncated order complex of generalized noncrossing partitions D. Armstrong and C. Krattenthaler Department of Mathematics, University of Miami, Coral Gables, Florida 33146,

More information

LARGE SCHRÖDER PATHS BY TYPES AND SYMMETRIC FUNCTIONS

LARGE SCHRÖDER PATHS BY TYPES AND SYMMETRIC FUNCTIONS Bull. Korean Math. Soc. 51 (2014), No. 4, pp. 1229 1240 http://dx.doi.org/10.4134/bkms.2014.51.4.1229 LARGE SCHRÖDER PATHS BY TYPES AND SYMMETRIC FUNCTIONS Su Hyung An, Sen-Peng Eu, and Sangwook Kim Abstract.

More information

Lecture Notes Introduction to Cluster Algebra

Lecture Notes Introduction to Cluster Algebra Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: May 16, 2017 5 Review of Root Systems In this section, let us have a brief introduction to root system and finite Lie type classification

More information

arxiv: v2 [math.co] 10 Sep 2010

arxiv: v2 [math.co] 10 Sep 2010 THE SHI ARRANGEMENT AND THE ISH ARRANGEMENT arxiv:1009.1655v2 [math.co] 10 Sep 2010 DREW ARMSTRONG AND BRENDON RHOADES Abstract. This paper is about two arrangements of hyperplanes. The first the Shi arrangement

More information

PARKING FUNCTIONS. Richard P. Stanley Department of Mathematics M.I.T Cambridge, MA

PARKING FUNCTIONS. Richard P. Stanley Department of Mathematics M.I.T Cambridge, MA PARKING FUNCTIONS Richard P. Stanley Department of Mathematics M.I.T. -75 Cambridge, MA 09 rstan@math.mit.edu http://www-math.mit.edu/~rstan Transparencies available at: http://www-math.mit.edu/~rstan/trans.html

More information

Problem 1. Let f n (x, y), n Z, be the sequence of rational functions in two variables x and y given by the initial conditions.

Problem 1. Let f n (x, y), n Z, be the sequence of rational functions in two variables x and y given by the initial conditions. 18.217 Problem Set (due Monday, December 03, 2018) Solve as many problems as you want. Turn in your favorite solutions. You can also solve and turn any other claims that were given in class without proofs,

More information

Tesler Matrices. Drew Armstrong. Bruce Saganfest Gainesville, March University of Miami armstrong

Tesler Matrices. Drew Armstrong. Bruce Saganfest Gainesville, March University of Miami  armstrong Tesler Matrices Drew Armstrong University of Miami www.math.miami.edu/ armstrong Bruce Saganfest Gainesville, March 2014 Plan 1. Tesler Matrices (with A. Garsia, J. Haglund, B. Rhoades and B. Sagan) 2.

More information

Clusters, Coxeter-sortable elements and noncrossing partitions

Clusters, Coxeter-sortable elements and noncrossing partitions Formal Power Series and Algebraic Combinatorics Séries Formelles et Combinatoire Algébrique San Diego, California 2006 Clusters, Coxeter-sortable elements and noncrossing partitions Extended abstract.

More information

DISSERTATION. Titel der Dissertation. q, t-fuß-catalan numbers for finite reflection groups. q, t-fuß-catalan Zahlen für endliche Spiegelungsgruppen

DISSERTATION. Titel der Dissertation. q, t-fuß-catalan numbers for finite reflection groups. q, t-fuß-catalan Zahlen für endliche Spiegelungsgruppen DISSERTATION Titel der Dissertation q, t-fuß-catalan numbers for finite reflection groups q, t-fuß-catalan Zahlen für endliche Spiegelungsgruppen Verfasser Christian Stump angestrebter akademischer Grad

More information

SHELLABILITY AND HIGHER COHEN-MACAULAY CONNECTIVITY OF GENERALIZED CLUSTER COMPLEXES

SHELLABILITY AND HIGHER COHEN-MACAULAY CONNECTIVITY OF GENERALIZED CLUSTER COMPLEXES ISRAEL JOURNAL OF MATHEMATICS 167 (2008), 177 191 DOI: 10.1007/s11856-008-1046-6 SHELLABILITY AND HIGHER COHEN-MACAULAY CONNECTIVITY OF GENERALIZED CLUSTER COMPLEXES BY Christos A. Athanasiadis Department

More information

FROM PARKING FUNCTIONS TO GELFAND PAIRS

FROM PARKING FUNCTIONS TO GELFAND PAIRS FROM PARKING FUNCTIONS TO GELFAND PAIRS KÜRŞAT AKER, MAHİR BİLEN CAN Abstract. A pair (G, K) of a group and its subgroup is called a Gelfand pair if the induced trivial representation of K on G is multiplicity

More information

Dynamical algebraic combinatorics

Dynamical algebraic combinatorics Dynamical algebraic combinatorics organized by James Propp, Tom Roby, Jessica Striker, and Nathan Williams Workshop Summary Topics of the Workshop This workshop centered around dynamical systems arising

More information

NONCROSSING PARTITIONS FOR THE GROUP D n

NONCROSSING PARTITIONS FOR THE GROUP D n NONCROSSING PARTITIONS FOR THE GROUP D n CHRISTOS A. ATHANASIADIS AND VICTOR REINER Dedicated to the memory of Rodica Simion Abstract. The poset of noncrossing partitions can be naturally defined for any

More information

Some Catalan Musings p. 1. Some Catalan Musings. Richard P. Stanley

Some Catalan Musings p. 1. Some Catalan Musings. Richard P. Stanley Some Catalan Musings p. 1 Some Catalan Musings Richard P. Stanley Some Catalan Musings p. 2 An OEIS entry A000108: 1,1,2,5,14,42,132,429,... Some Catalan Musings p. 2 An OEIS entry A000108: 1,1,2,5,14,42,132,429,...

More information

Rational Catalan Combinatorics

Rational Catalan Combinatorics Rational Catalan Combinatorics Eugene Gorsky UC Davis Bay Area Discrete Math Day October 17, 2015 Counting Dyck paths Catalan numbers The Catalan number is the number of Dyck paths, that is, lattice paths

More information

q,t-fuß Catalan numbers for finite reflection groups

q,t-fuß Catalan numbers for finite reflection groups J Algebr Comb (2010) 32: 67 97 DOI 10.1007/s10801-009-0205-0 q,t-fuß Catalan numbers for finite reflection groups Christian Stump Received: 12 January 2009 / Accepted: 30 September 2009 / Published online:

More information

Lecture 3: Tropicalizations of Cluster Algebras Examples David Speyer

Lecture 3: Tropicalizations of Cluster Algebras Examples David Speyer Lecture 3: Tropicalizations of Cluster Algebras Examples David Speyer Let A be a cluster algebra with B-matrix B. Let X be Spec A with all of the cluster variables inverted, and embed X into a torus by

More information

REPRESENTATIONS OF U(N) CLASSIFICATION BY HIGHEST WEIGHTS NOTES FOR MATH 261, FALL 2001

REPRESENTATIONS OF U(N) CLASSIFICATION BY HIGHEST WEIGHTS NOTES FOR MATH 261, FALL 2001 9 REPRESENTATIONS OF U(N) CLASSIFICATION BY HIGHEST WEIGHTS NOTES FOR MATH 261, FALL 21 ALLEN KNUTSON 1 WEIGHT DIAGRAMS OF -REPRESENTATIONS Let be an -dimensional torus, ie a group isomorphic to The we

More information

Longest element of a finite Coxeter group

Longest element of a finite Coxeter group Longest element of a finite Coxeter group September 10, 2015 Here we draw together some well-known properties of the (unique) longest element w in a finite Coxeter group W, with reference to theorems and

More information

DISSERTATION. Titel der Dissertation. q, t-fuß-catalan Zahlen für endliche Spiegelungsgruppen. Verfasser. Christian Stump

DISSERTATION. Titel der Dissertation. q, t-fuß-catalan Zahlen für endliche Spiegelungsgruppen. Verfasser. Christian Stump DISSERTATION Titel der Dissertation q, t-fuß-catalan numbers for finite reflection groups q, t-fuß-catalan Zahlen für endliche Spiegelungsgruppen Verfasser Christian Stump angestrebter akademischer Grad

More information

A BIJECTION BETWEEN NONCROSSING AND NONNESTING PARTITIONS FOR CLASSICAL REFLECTION GROUPS

A BIJECTION BETWEEN NONCROSSING AND NONNESTING PARTITIONS FOR CLASSICAL REFLECTION GROUPS A BIJECTION BETWEEN NONCROSSING AND NONNESTING PARTITIONS FOR CLASSICAL REFLECTION GROUPS ALEX FINK AND BENJAMIN IRIARTE GIRALDO Abstract. We present an elementary type preserving bijection between noncrossing

More information

Again we return to a row of 1 s! Furthermore, all the entries are positive integers.

Again we return to a row of 1 s! Furthermore, all the entries are positive integers. Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Updated: April 4, 207 Examples. Conway-Coxeter frieze pattern Frieze is the wide central section part of an entablature, often seen in Greek temples,

More information

A connection between number theory and linear algebra

A connection between number theory and linear algebra A connection between number theory and linear algebra Mark Steinberger Contents 1. Some basics 1 2. Rational canonical form 2 3. Prime factorization in F[x] 4 4. Units and order 5 5. Finite fields 7 6.

More information

arxiv: v1 [math.co] 24 Jul 2013

arxiv: v1 [math.co] 24 Jul 2013 BIJECTIONS FOR THE SHI AND ISH ARRANGEMENTS EMILY LEVEN, BRENDON RHOADES, AND ANDREW TIMOTHY WILSON arxiv:07.65v [math.co] 4 Jul 0 Abstract. The Shi hyperplane arrangement Shi(n) was introduced by Shi

More information

Some notes on Coxeter groups

Some notes on Coxeter groups Some notes on Coxeter groups Brooks Roberts November 28, 2017 CONTENTS 1 Contents 1 Sources 2 2 Reflections 3 3 The orthogonal group 7 4 Finite subgroups in two dimensions 9 5 Finite subgroups in three

More information

On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem

On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem Bertram Kostant, MIT Conference on Representations of Reductive Groups Salt Lake City, Utah July 10, 2013

More information

Standard Young Tableaux Old and New

Standard Young Tableaux Old and New Standard Young Tableaux Old and New Ron Adin and Yuval Roichman Department of Mathematics Bar-Ilan University Workshop on Group Theory in Memory of David Chillag Technion, Haifa, Oct. 14 1 2 4 3 5 7 6

More information

Root system chip-firing

Root system chip-firing Root system chip-firing PhD Thesis Defense Sam Hopkins Massachusetts Institute of Technology April 27th, 2018 Includes joint work with Pavel Galashin, Thomas McConville, Alexander Postnikov, and James

More information

ENUMERATION OF TREES AND ONE AMAZING REPRESENTATION OF THE SYMMETRIC GROUP. Igor Pak Harvard University

ENUMERATION OF TREES AND ONE AMAZING REPRESENTATION OF THE SYMMETRIC GROUP. Igor Pak Harvard University ENUMERATION OF TREES AND ONE AMAZING REPRESENTATION OF THE SYMMETRIC GROUP Igor Pak Harvard University E-mail: pak@math.harvard.edu Alexander Postnikov Massachusetts Institute of Technology E-mail: apost@math.mit.edu

More information

RCC. Drew Armstrong. FPSAC 2017, Queen Mary, London. University of Miami armstrong

RCC. Drew Armstrong. FPSAC 2017, Queen Mary, London. University of Miami   armstrong RCC Drew Armstrong University of Miami www.math.miami.edu/ armstrong FPSAC 2017, Queen Mary, London Outline of the Talk 1. The Frobenius Coin Problem 2. Rational Dyck Paths 3. Core Partitions 4. The Double

More information

Some Catalan Musings p. 1. Some Catalan Musings. Richard P. Stanley

Some Catalan Musings p. 1. Some Catalan Musings. Richard P. Stanley Some Catalan Musings p. 1 Some Catalan Musings Richard P. Stanley Some Catalan Musings p. 2 An OEIS entry A000108: 1,1,2,5,14,42,132,429,... Some Catalan Musings p. 2 An OEIS entry A000108: 1,1,2,5,14,42,132,429,...

More information

16.2. Definition. Let N be the set of all nilpotent elements in g. Define N

16.2. Definition. Let N be the set of all nilpotent elements in g. Define N 74 16. Lecture 16: Springer Representations 16.1. The flag manifold. Let G = SL n (C). It acts transitively on the set F of complete flags 0 F 1 F n 1 C n and the stabilizer of the standard flag is the

More information

ENUMERATION OF CONNECTED CATALAN OBJECTS BY TYPE. 1. Introduction

ENUMERATION OF CONNECTED CATALAN OBJECTS BY TYPE. 1. Introduction ENUMERATION OF CONNECTED CATALAN OBJECTS BY TYPE BRENDON RHOADES Abstract. Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects

More information

Dual braid monoids and Koszulity

Dual braid monoids and Koszulity Dual braid monoids and Koszulity Phillippe Nadeau (CNRS, Univ. Lyon 1) SLC 69, Strobl, September 10th 2012 Goal Investigate the combinatorics of a certain graded algebra associated to a Coxeter system,

More information

L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that

L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive

More information

Workshop on B-stable ideals and nilpotent orbits

Workshop on B-stable ideals and nilpotent orbits Workshop on B-stable ideals and nilpotent orbits October 8-12, Roma, Italia Opening October 8, 10:15 Schedule of Talks Mon: 10:30-11:30 11:30-12 12-13 15-16 16-17 Kumar Coffee Break Möseneder Panyushev

More information

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

More information

Abstract Algebra Study Sheet

Abstract Algebra Study Sheet Abstract Algebra Study Sheet This study sheet should serve as a guide to which sections of Artin will be most relevant to the final exam. When you study, you may find it productive to prioritize the definitions,

More information

Actions and Identities on Set Partitions

Actions and Identities on Set Partitions Actions and Identities on Set Partitions The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Marberg,

More information

REPRESENTATIONS OF S n AND GL(n, C)

REPRESENTATIONS OF S n AND GL(n, C) REPRESENTATIONS OF S n AND GL(n, C) SEAN MCAFEE 1 outline For a given finite group G, we have that the number of irreducible representations of G is equal to the number of conjugacy classes of G Although

More information

The (q, t)-catalan Numbers and the Space of Diagonal Harmonics. James Haglund. University of Pennsylvania

The (q, t)-catalan Numbers and the Space of Diagonal Harmonics. James Haglund. University of Pennsylvania The (q, t)-catalan Numbers and the Space of Diagonal Harmonics James Haglund University of Pennsylvania Outline Intro to q-analogues inv and maj q-catalan Numbers MacMahon s q-analogue The Carlitz-Riordan

More information

Symmetries and Polynomials

Symmetries and Polynomials Symmetries and Polynomials Aaron Landesman and Apurva Nakade June 30, 2018 Introduction In this class we ll learn how to solve a cubic. We ll also sketch how to solve a quartic. We ll explore the connections

More information

1 Fields and vector spaces

1 Fields and vector spaces 1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary

More information

THE SEMISIMPLE SUBALGEBRAS OF EXCEPTIONAL LIE ALGEBRAS

THE SEMISIMPLE SUBALGEBRAS OF EXCEPTIONAL LIE ALGEBRAS Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 67 (2006) 2006, Pages 225 259 S 0077-1554(06)00156-7 Article electronically published on December 27, 2006 THE SEMISIMPLE SUBALGEBRAS OF EXCEPTIONAL

More information

Rational Parking Functions

Rational Parking Functions Rational Parking Functions Drew Armstrong et al. University of Miami www.math.miami.edu/ armstrong December 9, 2012 Rational Catalan Numbers CONVENTION Given x Q \ [ 1, 0], there exist unique coprime (a,

More information

Citation Osaka Journal of Mathematics. 43(2)

Citation Osaka Journal of Mathematics. 43(2) TitleIrreducible representations of the Author(s) Kosuda, Masashi Citation Osaka Journal of Mathematics. 43(2) Issue 2006-06 Date Text Version publisher URL http://hdl.handle.net/094/0396 DOI Rights Osaka

More information

(Ref: Schensted Part II) If we have an arbitrary tensor with k indices W i 1,,i k. we can act on it 1 2 k with a permutation P = = w ia,i b,,i l

(Ref: Schensted Part II) If we have an arbitrary tensor with k indices W i 1,,i k. we can act on it 1 2 k with a permutation P = = w ia,i b,,i l Chapter S k and Tensor Representations Ref: Schensted art II) If we have an arbitrary tensor with k indices W i,,i k ) we can act on it k with a permutation = so a b l w) i,i,,i k = w ia,i b,,i l. Consider

More information

The Gaussian coefficient revisited

The Gaussian coefficient revisited The Gaussian coefficient revisited Richard EHRENBORG and Margaret A. READDY Abstract We give new -(1+)-analogue of the Gaussian coefficient, also now as the -binomial which, lie the original -binomial

More information

Topics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem

Topics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem Topics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem 1 Fourier Analysis, a review We ll begin with a short review of simple facts about Fourier analysis, before going on to interpret

More information

Proof of the Broué Malle Rouquier Conjecture in Characteristic Zero (After I. Losev and I. Marin G. Pfeiffer)

Proof of the Broué Malle Rouquier Conjecture in Characteristic Zero (After I. Losev and I. Marin G. Pfeiffer) Arnold Math J. DOI 10.1007/s40598-017-0069-7 RESEARCH EXPOSITION Proof of the Broué Malle Rouquier Conjecture in Characteristic Zero (After I. Losev and I. Marin G. Pfeiffer) Pavel Etingof 1 Received:

More information

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1

More information

REPRESENTATION THEORY OF S n

REPRESENTATION THEORY OF S n REPRESENTATION THEORY OF S n EVAN JENKINS Abstract. These are notes from three lectures given in MATH 26700, Introduction to Representation Theory of Finite Groups, at the University of Chicago in November

More information

Tangent spaces, normals and extrema

Tangent spaces, normals and extrema Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent

More information

Rigidity of Rings and Invariants of the Weyl Algebra I

Rigidity of Rings and Invariants of the Weyl Algebra I Rigidity of Rings and Invariants of the Weyl Algebra I João Fernando Schwarz Universidade de São Paulo 15 de março de 2018 Introducing the notion of rigidity All base fields k will have char = 0. All automorphisms

More information

Notes 10: Consequences of Eli Cartan s theorem.

Notes 10: Consequences of Eli Cartan s theorem. Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation

More information

MULTI-ORDERED POSETS. Lisa Bishop Department of Mathematics, Occidental College, Los Angeles, CA 90041, United States.

MULTI-ORDERED POSETS. Lisa Bishop Department of Mathematics, Occidental College, Los Angeles, CA 90041, United States. INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A06 MULTI-ORDERED POSETS Lisa Bishop Department of Mathematics, Occidental College, Los Angeles, CA 90041, United States lbishop@oxy.edu

More information

(Affine) generalized associahedra, root systems, and cluster algebras

(Affine) generalized associahedra, root systems, and cluster algebras (Affine) generalized associahedra, root systems, and cluster algebras Salvatore Stella (joint with Nathan Reading) INdAM - Marie Curie Actions fellow Dipartimento G. Castelnuovo Università di Roma La Sapienza

More information

Tropicalizations of Positive Parts of Cluster Algebras The conjectures of Fock and Goncharov David Speyer

Tropicalizations of Positive Parts of Cluster Algebras The conjectures of Fock and Goncharov David Speyer Tropicalizations of Positive Parts of Cluster Algebras The conjectures of Fock and Goncharov David Speyer This talk is based on arxiv:math/0311245, section 4. We saw before that tropicalizations look like

More information

Modular representations of symmetric groups: An Overview

Modular representations of symmetric groups: An Overview Modular representations of symmetric groups: An Overview Bhama Srinivasan University of Illinois at Chicago Regina, May 2012 Bhama Srinivasan (University of Illinois at Chicago) Modular Representations

More information

Definition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O).

Definition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O). 9. Calogero-Moser spaces 9.1. Hamiltonian reduction along an orbit. Let M be an affine algebraic variety and G a reductive algebraic group. Suppose M is Poisson and the action of G preserves the Poisson

More information

11. Periodicity of Klein polyhedra (28 June 2011) Algebraic irrationalities and periodicity of sails. Let A GL(n + 1, R) be an operator all of

11. Periodicity of Klein polyhedra (28 June 2011) Algebraic irrationalities and periodicity of sails. Let A GL(n + 1, R) be an operator all of 11. Periodicity of Klein polyhedra (28 June 2011) 11.1. lgebraic irrationalities and periodicity of sails. Let GL(n + 1, R) be an operator all of whose eigenvalues are real and distinct. Let us take the

More information

Representation Theory

Representation Theory Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 1, Section II 19I 93 (a) Define the derived subgroup, G, of a finite group G. Show that if χ is a linear character

More information

2-FRIEZE PATTERNS AND THE CLUSTER STRUCTURE OF THE SPACE OF POLYGONS

2-FRIEZE PATTERNS AND THE CLUSTER STRUCTURE OF THE SPACE OF POLYGONS -FRIEZE PATTERNS AND THE CLUSTER STRUCTURE OF THE SPACE OF POLYGONS SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO, AND SERGE TABACHNIKOV Abstract We study the space of -frieze patterns generalizing that of the

More information

Jessica Striker. November 10, 2015

Jessica Striker. November 10, 2015 Resonance in dynamical algebraic combinatorics Jessica Striker joint work with Kevin Dilks (NDSU) and Oliver Pechenik (UIUC) North Dakota State University November 10, 2015 J. Striker (NDSU) Resonance

More information

Patterns in Standard Young Tableaux

Patterns in Standard Young Tableaux Patterns in Standard Young Tableaux Sara Billey University of Washington Slides: math.washington.edu/ billey/talks Based on joint work with: Matjaž Konvalinka and Joshua Swanson 6th Encuentro Colombiano

More information

Combinatorics for algebraic geometers

Combinatorics for algebraic geometers Combinatorics for algebraic geometers Calculations in enumerative geometry Maria Monks March 17, 214 Motivation Enumerative geometry In the late 18 s, Hermann Schubert investigated problems in what is

More information

Finding parking when not commuting. Jon McCammond U.C. Santa Barbara

Finding parking when not commuting. Jon McCammond U.C. Santa Barbara Finding parking when not commuting 7 6 8 1 2 3 5 4 {{1,4,5}, {2,3}, {6,8}, {7}} Jon McCammond U.C. Santa Barbara 1 A common structure The main goal of this talk will be to introduce you to a mathematical

More information

arxiv:math/ v1 [math.co] 10 Nov 1998

arxiv:math/ v1 [math.co] 10 Nov 1998 A self-dual poset on objects counted by the Catalan numbers arxiv:math/9811067v1 [math.co] 10 Nov 1998 Miklós Bóna School of Mathematics Institute for Advanced Study Princeton, NJ 08540 April 11, 2017

More information

Counting matrices over finite fields

Counting matrices over finite fields Counting matrices over finite fields Steven Sam Massachusetts Institute of Technology September 30, 2011 1/19 Invertible matrices F q is a finite field with q = p r elements. [n] = 1 qn 1 q = qn 1 +q n

More information

Outline. 1 Geometry and Commutative Algebra. 2 Singularities and Resolutions. 3 Noncommutative Algebra and Deformations. 4 Representation Theory

Outline. 1 Geometry and Commutative Algebra. 2 Singularities and Resolutions. 3 Noncommutative Algebra and Deformations. 4 Representation Theory Outline Geometry, noncommutative algebra and representations Iain Gordon http://www.maths.ed.ac.uk/ igordon/ University of Edinburgh 16th December 2006 1 2 3 4 1 Iain Gordon Geometry, noncommutative algebra

More information

3 Galois Theory. 3.1 Definitions and Examples

3 Galois Theory. 3.1 Definitions and Examples 3 Galois Theory 3.1 Definitions and Examples This section of notes roughly follows Section 14.1 in Dummit and Foote. Let F be a field and let f (x) 2 F[x]. In the previous chapter, we proved that there

More information

Resonance in orbits of plane partitions and increasing tableaux. Jessica Striker North Dakota State University

Resonance in orbits of plane partitions and increasing tableaux. Jessica Striker North Dakota State University Resonance in orbits of plane partitions and increasing tableaux Jessica Striker North Dakota State University joint work with Kevin Dilks (NDSU) and Oliver Pechenik (UIUC) February 19, 2016 J. Striker

More information

SEPARABLE EXTENSIONS OF DEGREE p IN CHARACTERISTIC p; FAILURE OF HERMITE S THEOREM IN CHARACTERISTIC p

SEPARABLE EXTENSIONS OF DEGREE p IN CHARACTERISTIC p; FAILURE OF HERMITE S THEOREM IN CHARACTERISTIC p SEPARABLE EXTENSIONS OF DEGREE p IN CHARACTERISTIC p; FAILURE OF HERMITE S THEOREM IN CHARACTERISTIC p JIM STANKEWICZ 1. Separable Field Extensions of degree p Exercise: Let K be a field of characteristic

More information

H(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )).

H(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )). 92 19. Perverse sheaves on the affine Grassmannian 19.1. Spherical Hecke algebra. The Hecke algebra H(G(Q p )//G(Z p )) resp. H(G(F q ((T ))//G(F q [[T ]])) etc. of locally constant compactly supported

More information

The structure of euclidean Artin groups

The structure of euclidean Artin groups The structure of euclidean Artin groups Jon McCammond UC Santa Barbara Cortona Sept 2014 Coxeter groups The spherical and euclidean Coxeter groups are reflection groups that act geometrically on spheres

More information

arxiv: v1 [math.rt] 14 Nov 2007

arxiv: v1 [math.rt] 14 Nov 2007 arxiv:0711.2128v1 [math.rt] 14 Nov 2007 SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS: CORRIGENDA AND ADDENDA ALEXANDER PREMET J. C. Jantzen informed me that the proof

More information

Topics in linear algebra

Topics in linear algebra Chapter 6 Topics in linear algebra 6.1 Change of basis I want to remind you of one of the basic ideas in linear algebra: change of basis. Let F be a field, V and W be finite dimensional vector spaces over

More information

Philip Puente. Statement of Research and Scholarly Interests 1 Areas of Interest

Philip Puente. Statement of Research and Scholarly Interests 1 Areas of Interest Philip Puente Statement of Research and Scholarly Interests 1 Areas of Interest I work on questions on reflection groups at the intersection of algebra, geometry, and topology. My current research uses

More information

Canonical systems of basic invariants for unitary reflection groups

Canonical systems of basic invariants for unitary reflection groups Canonical systems of basic invariants for unitary reflection groups Norihiro Nakashima, Hiroaki Terao and Shuhei Tsujie Abstract It has been known that there exists a canonical system for every finite

More information

Cluster Variables and Perfect Matchings of Subgraphs of the dp 3 Lattice

Cluster Variables and Perfect Matchings of Subgraphs of the dp 3 Lattice Cluster Variables and Perfect Matchings of Subgraphs of the dp Lattice Sicong Zhang October 8, 0 Abstract We give a combinatorial intepretation of cluster variables of a specific cluster algebra under

More information